Applicable Algebra in Engineering Communication and Computing最新文献

Illustrated by a problem on paint pots that is easy to understand but hard to solve, we investigate whether particular monoids have the property of common right multiples. As one result we characterize generalized braid monoids represented by undirected graphs, being a subclass of Artin–Tits monoids. Stated in other words, we investigate to which graphs the old Garside result stating that braid monoids have the property of common right multiples, generalizes. This characterization also follows from old results on Coxeter groups and the connection between finiteness of Coxeter groups and common right multiples in Artin–Tits monoids. However, our independent presentation is self-contained up to some basic knowledge of rewriting, and also applies to monoids beyond the Artin–Tits format. The main new contribution is a technique to prove that the property of common right multiples does not hold, by finding a particular model, in our examples all being finite.

Cyclic codes are an important subclass of linear codes. In this paper, we investigate the construction of quinary cyclic codes with parameters ([5^{m}-1, 5^{m}-2m-2, 4]) and eight new classes optimal quinary cyclic codes of form ({mathcal {C}}_{(1,e,s)}) are presented by discussing the solutions of certain equations over ({mathbb {F}}_{5^{m}}).

In this paper we introduce and analyze a discrete SIS epidemic model for a homogeneous population. As a discretization method the strictly positive scheme was chosen. The presented model is built from its continuous counterpart known from literature. We firstly present basic properties of the system. Later we discuss local stability of stationary states and global stability for the disease-free stationary state. The results for this state are expressed with the use of the basic reproduction number. The main conclusion from our work is that conditions for stability of the stationary states do not depend on the step size of the discretization method. This fact stays in contrary to other discrete models analyzed in our previous papers. Theoretical results are accomplished with numerical simulations.

Algebraic Cryptanalysis is a widely used technique that tackles the problem of breaking ciphers mainly relying on the ability to express a cryptosystem as a solvable polynomial system. Each output bit/word can be expressed as a polynomial equation in the cipher’s inputs—namely the key and the plaintext or the initialisation vector bits/words. A part of research in this area consists in finding suitable algebraic structures where polynomial systems can be effectively solved, e.g., by computing Gröbner bases. In 2009, Dinur and Shamir proposed the cube attack, a chosen plaintext algebraic cryptanalysis technique for the offline acquisition of an equivalent system by means of monomial reduction; interpolation on cubes in the space of variables enables retrieving a linear polynomial system, hence making it exploitable in the online phase to recover the secret key. Since its introduction, this attack has received both many criticisms and endorsements from the crypto community; this work aims at providing, under a unified notation, a complete state-of-the-art review of recent developments by categorising contributions in five classes. We conclude the work with an in-depth description of the kite attack framework, a cipher-independent tool that implements cube attacks on GPUs. Mickey2.0 is adopted as a showcase.

There are various metrics for researching error-correcting codes. Especially, high-density data storage system gives the existence of inconsistency for the reading and writing process. The symbol-pair metric is motivated for outputs that have overlapping pairs of symbols in a certain channel. The Rosenbloom–Tsfasman (RT) metric is introduced since there exists a problem that is related to transmission over several parallel communication channels with some channels not available for the transmission. In this paper, we determine the minimum symbol-pair weight and RT weight of repeated-root cyclic codes over (mathfrak R=mathbb {F}_{p^m}[u]/langle u^4rangle ) of length (n=p^k). For the determination, we explicitly present third torsional degree for all different types of cyclic codes over (mathfrak R) of length n.

The strongly annihilating-ideal graph (textrm{SAG}(R)) of a commutative unital ring R is a simple graph whose vertices are non-zero ideals of R with non-zero annihilator and there exists an edge between two distinct vertices if and only if each of them has a non-zero intersection with annihilator of the other one. In this paper, we compute twin-free clique number of (textrm{SAG}(R)) and as an application strong metric dimension of (textrm{SAG}(R)) is given. Moreover, we investigate the structures of strong resolving sets in (textrm{SAG}(R)) to find forcing strong metric dimension in (textrm{SAG}(R)).

The task of recognizing an algebraic surface from a single apparent contour can be reduced to the recovering of a homogeneous equation in four variables from its discriminant. In this paper, we use the fact that Darboux cyclides have a singularity along the absolute conic in order to recognize them up to Euclidean similarity transformations.

Sequences of consecutive Legendre and Jacobi symbols as pseudorandom bit generators were proposed for cryptographic use in 1988. Major interest has been shown towards pseudorandom functions (PRF) recently, based on the Legendre and power residue symbols, due to their efficiency in the multi-party setting. The security of these PRFs is not known to be reducible to standard cryptographic assumptions. In this work, we show that key-recovery attacks against the Legendre PRF are equivalent to solving a specific family of multivariate quadratic (MQ) equation system over a finite prime field. This new perspective sheds some light on the complexity of key-recovery attacks against the Legendre PRF. We conduct algebraic cryptanalysis on the resulting MQ instance. We show that the currently known techniques and attacks fall short in solving these sparse quadratic equation systems. Furthermore, we build novel cryptographic applications of the Legendre PRF, e.g., verifiable random function and (verifiable) oblivious (programmable) PRFs.

In this paper, we compute sequence covering arrays (SCAs), which are arrays, consisting of sequences, such that all subsequences with pairwise different entries of some length are covered, via a novel approach based on commutative algebra and symbolic computation. Hereby, we provide various algebraic models being capable to characterize possibly small sets of permutations collectively containing particular shorter subsequences. These models take the form of multivariate polynomial systems of equations and are then processed via supercomputing by a Gröbner Basis solver in order to compute solutions from them. If the variety is not empty, i.e. the Gröbner basis is non-trivial, then each point in the computed variety can be transformed to a SCA. In our experiments, we observed varying computational performance depending on the chosen model, while all of them exhibited scalability issues. Additionally and for comparison, we give new SAT descriptions modelling SCAs. By employing a SAT solver on our provided SAT models, we are able to provide upper bounds, one of which is best among literature results. Lastly, we adapt our SAT approach to answer a question posed by Yuster (Des Codes Cryptogr 88(3):585–593, 2020). As a result, we find a characterization of the dimensions of all perfect SCAs with coverage multiplicity two of strength three.